empty family

Given any set XX, there is a unique empty family of elements of XX. Formally, this is given by the empty function to XX, the unique function from to XX from the empty set. As the empty set is finite and (a fortiori) countable, this empty family counts as a list and a sequence; in such a guise it is known as the empty list or the empty sequence.

When treating it as an element of the free monoid on XX, the empty list may be written ()(), **, or ϵ\epsilon, perhaps with a subscript XX if desired.

Similarly, we have the notions of the empty family of elements of a preset or other notion of type, the empty family of objects and the empty family of morphisms of a given category, and more generally the empty family of whatever you want.

Revised on August 19, 2010 18:05:38 by Toby Bartels (